RD Sharma Solutions for Class 8 Math Chapter 7 - Factorization (Part-6) | RD Sharma Solutions for Class 8 Mathematics PDF Download

Question 1: Factorize each of the following algebraic expression:
4x² + 12xy +9y²
Answer 1: 4x2+12xy+9y2 
=(2x)2+2×2x×3y+(3y)2 
=(2x+3y)2 
=(2x+3y)(2x+3y) 
Question 2: Factorize each of the following algebraic expression:
9a² − 24ab + 16b²
Answer 2: 9a2−24ab+16b2 
=(3a)2−2×3a×4b+(4b)2 
=(3a−4b)2 
=(3a−4b)(3a−4b) 
Question 3: Factorize each of the following algebraic expression:
p²q² − 6pqr + 9r²
Answer 3:
p²q²−6pqr+9r² 
=(pq)2−2×pq×3r+(3r)2 
=(pq−3r)2 
=(pq−3r)(pq−3r) 
Question 4: Factorize each of the following algebraic expression:
36a² + 36a + 9
Answer 4: 36a2+36a+9 
=9(4a2+4a+1) 
=9{(2a)2+2×2a×1+12} 
=9(2a+1)2 
=9(2a+1)(2a+1) 
Question 5: Factorize each of the following algebraic expression:
a² + 2ab + b² − 16
Answer 5: a2+2ab+b2−16 
=a2+2×a×b+b2−16 
=(a+b)2−42 
=(a+b−4)(a+b+4) 
Question 6: Factorize each of the following algebraic expression:
9z² − x² + 4xy − 4y²
Answer 6: 9z2−x2+4xy−4y2 
=9z2−(x2−4xy+4y2) 
=9z2−[x2−2×x×2y+(2y)2] 
=(3z)2−(x−2y)2 
=[3z−(x−2y)][3z+(x−2y)]
=(3z−x+2y)(3z+x−2y)=(x−2y+3z)(−x+2y+3z)
Question 7: Factorize each of the following algebraic expression:
9a⁴ − 24a²b² + 16b⁴ − 256
Answer 7: 9a4−24a2b2+16b4−256 
=(9a4−24a2b2+16b4)−256 
=[(3a2)2−2×3a2×4b2+(4b2)2]−162 
=(3a2−4b2)2−162 
=[(3a2−4b2)−16][(3a2−4b2)+16] 
=(3a2−4b2−16)(3a2−4b2+16) 
Question 8: Factorize each of the following algebraic expression:
16 − a⁶ + 4a³b³ − 4b⁶
Answer 8: 16−a6+4a3b3−4b6 
=16−(a6−4a3b3+4b6) 
=42−[(a3)2−2×a3×2b3+(2b3)2] 
=42−(a3−2b3)2 
=[4−(a3−2b3)][4+(a3−2b3)] 
=(4−a3+2b3)(4+a3−2b3) 
=(a3−2b3+4)(−a3+2b3+4) 
Question 9: Factorize each of the following algebraic expression:
a² − 2ab + b² − c²
Answer 9: a2−2ab+b2−c2 
=(a2−2ab+b2)−c2 
=(a2−2×a×b+b2)−c2 
=(a−b)2−c2 
=[(a−b)−c][(a−b)+c] 
=(a−b−c)(a−b+c)
Question 10: Factorize each of the following algebraic expression:
x² + 2x + 1 − 9y²
Answer 10: x2+2x+1−9y2 
=(x2+2x+1)−9y2 
=(x2+2×x×1+1)−9y2 
=(x+1)2−(3y)2 
=[(x+1)−3y][(x+1)+3y] 
=(x+1−3y)(x+1+3y) 
=(x+3y+1)(x−3y+1) 
Question 11: Factorize each of the following algebraic expression:
a² + 4ab + 3b²
Answer 11: a2+4ab+3b2 
=a2+4ab+4b2−b2 
=[a2+2×a×2b+(2b)2]−b2 
=(a+2b)2−b2 
=[(a+2b)−b][(a+2b)+b] 
=(a+2b−b)(a+2b+b) 
=(a+b)(a+3b) 
Question 12: Factorize each of the following algebraic expression:
96 − 4x − x²
Answer 12: 96−4x−x2 
=100−4−4x−x2 
=100−(x2+4x+4) 
=100−(x2+2×x×2+22) 
=102−(x+2)2 
=[10−(x+2)][10+(x+2)] 
=(10−x−2)(10+x+2) 
=(8−x)(12+x) 
=(x+12)(−x+8) 
Question 13: Factorize each of the following algebraic expression:
a⁴ + 3a² +4
Answer 13: a4+3a2+4 
=a4+4a2−a2+4 
=(a4+4a2+4)−a2 
=[(a2)2+2×a2×2+22]−a2 
=(a2+2)2−a2 
=[(a2+2)−a][(a2+2)+a] 
=(a2−a+2)((a2+a+2) 
Question 14: Factorize each of the following algebraic expression:
4x⁴ + 1
Answer 14:
4x⁴+1
=4x4+4x2+1−4x2 
=[(2x2)2+2×2x2×1+1]−4x2 
=(2x2+1)2−(2x)2 
=[(2x2+1)−2x][(2x2+1)+2x] 
=(2x2−2x+1)(2x2+2x+1) 
Question 15: Factorize each of the following algebraic expression:
4x⁴ + y⁴ 
Answer 15: 4x4+y4 
=4x4+4x2y2+y4−4x2y2 
=[(2x2)2+2×2x2×y+(y2)2]−(2xy)2 
=(2x2+y2)2−(2xy)2 
=[(2x2+y2)−2xy][(2x2+y2)+2xy] 
=(2x2−2xy+y2)(2x2+2xy+y2) 
Question 16: Factorize each of the following algebraic expression:
(x + 2)² − 6(x + 2) + 9
Answer 16: (x+2)2−6(x+2)+9 
=(x+2)2−2×(x+2)×3+32 
=[(x+2)−3]2 
=(x+2−3)2 
=(x−1)2 
=(x−1)(x−1)
Question 17: Factorize each of the following algebraic expression:
25 − p² − q² − 2pq
Answer 17: 25−p2−q2−2pq 
=25−(p2+2pq+q2) 
=52−(p2+2×p×q+q2) 
=52−(p+q)2 
=[5−(p+q)][5+(p+q)] 
=(5−p−q)(5+p+q) 
=−(p+q−5)(p+q+5) 
Question 18: Factorize each of the following algebraic expression:
x² + 9y² − 6xy − 25a²
Answer 18:
 x2+9y2−6xy−25a2 
=(x2−6xy+9y2)−25a2 
=[x2−2×x×3y+(3y)2]−25a2 
=(x−3y)2−(5a)2 
=[(x−3y)−5a][(x−3y)+5a] 
=(x−3y−5a)(x−3y+5a) 
Question 19: Factorize each of the following algebraic expression:
49 − a² + 8ab − 16b²
Answer 19: 49−a2+8ab−16b2 
=49−(a2−8ab+16b2) 
=49−[a2−2×a×4b+(4b)2] 
=72−(a−4b)2 
=[7−(a−4b)][7+(a−4b)] 
=(7−a+4b)(7+a−4b) 
=−(a−4b−7)(a−4b+7) 
=−(a−4b+7)(a−4b−7) 
Question 20: Factorize each of the following algebraic expression:
a² − 8ab + 16b² − 25c²
Answer 20: a2−8ab+16b2−25c2 
=(a2−8ab+16b2)−25c2 
=[a2−2×a×4b+(4b)2]−25c2 
=(a−4b)2−(5c)2 
=[(a−4b)−5c][(a−4b)+5c] 
=(a−4b−5c)(a−4b+5c) 
Question 21: Factorize each of the following algebraic expression:
x² − y² + 6y − 9
Answer 21: x2−y2+6y−9 
=x2−(y2−6y+9) 
=x2−(y2−2×y×3+32) 
=x2−(y−3)2 
=[x−(y−3)][x+(y−3)] 
=(x−y+3)(x+y−3) 
Question 22: Factorize each of the following algebraic expression:
25x² − 10x + 1 − 36y²
Answer 22: 25x2−10x+1−36y2 
=(25x2−10x+1)−36y2 
=[(5x)2−2×5x×1+1]−36y2 
=(5x−1)2−(6y)2 
=[(5x−1)−6y][(5x−1)+6y] 
=(5x−1−6y)(5x−1+6y) 
=(5x−6y−1)(5x+6y−1) 
Question 23: Factorize each of the following algebraic expression:
a² − b² + 2bc − c²
Answer 23: a2−b2+2bc−c2 
=a2−(b2−2bc+c2) 
=a2−(b2−2×b×c+c2) 
=a2−(b−c)2 
=[(a−(b−c)][(a+(b−c)] 
=(a−b+c)(a+b−c) 
Question 24: Factorize each of the following algebraic expression:
a² + 2ab + b² − c²
Answer 24: a2+2ab+b2−c2
=(a2+2ab+b2)−c2 
=(a2+2×a×b+b2)−c2 
=(a+b)2−c2 
=[(a+b)−c][(a+b)+c] 
=(a+b−c)(a+b+c) 
Question 25: Factorize each of the following algebraic expression:
49 − x² − y² + 2xy
Answer 25: 49−x2−y2+2xy 
=49−(x2−2xy+y2) 
=49−(x2−2×x×y+y2) 
=72−(x−y)2 
=[7−(x−y)][7+(x−y)] 
=(7−x+y)(7+x−y) 
=(x−y+7)(y−x+7) 
Question 26: Factorize each of the following algebraic expression:
a² + 4b² − 4ab − 4c²
Answer 26: a2+4b2−4ab−4c2 
=(a2−4ab+4b2)−4c2 
=[a2−2×a×2b+(2b)2]−4c2 
=(a−2b)2−(2c)2 
=[(a−2b)−2c][(a−2b)+2c] 
=(a−2b−2c)(a−2b+2c) 
Question 27: Factorize each of the following algebraic expression:
x² − y² − 4xz + 4z²
Answer 27: x2−y2−4xz+4z2 
=(x2−4xz+4z2)−y2 
=[x2−2×x×2z+(2z)2]−y2 
=(x−2z)2−y2 
=[(x−2z)−y][(x−2z)+y] 
=(x−2z−y)(x−2z+y) 
=(x+y−2z)(x−y−2z)